Fejér--Riesz factorization for positive noncommutative trigonometric polynomials

Igor Klep; Jacob Levenson; Scott McCullough

Paper

Fejér--Riesz factorization for positive noncommutative trigonometric polynomials

Abstract

We prove a Fejér-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on

, where

is either the free semigroup

or the free product group

, and

is a discrete group. More precisely, using the shortlex order, if

has degree at most

in the

variables and is uniformly strictly positive on all unitary representations of

, then

with

analytic and of

-degree at most

; this degree bound is optimal, and strict positivity is essential. As an application, we obtain degree-bounded sums-of-squares certificates for Bell-type inequalities in

from quantum information theory. In the special case

we recover, in the matrix-valued setting, the classical commutative multivariable Fejér-Riesz factorization. For trivial

we obtain a ``perfect'' group-algebra Positivstellensatz on

that does not require strict positivity; this result is sharp, as demonstrated by counterexamples in

and

. To establish our main results two novel ingredients of independent interest are developed: (a) a positive-semidefinite Parrott theorem with entries given by functions on a group; and (b) solutions to positive semidefinite matrix completion problems for

or the free product group

indexed by words in

of length